& Frisch (1980), by a shift of origin. It offers the advantage of the symmetric Fourier representation (1 2 ) . Early-time behaviour of the inviscid flowThe symmetries of the TG vortex are listed in appendix A. Here we emphasize only those that help to visualize the qualitative features of the flow and those that may be important in making this flow atypical of general three-dimensional flow. First, for all times, no fluid crosses any of the boundaries x , y or z = nn, where n is an integer.Therefore the flow can be visualized as flow in the box 0 < x , y , z < n with impermeable stress-free faces. I n the following discussion, the region 0 < x, y , z < n is termed the impermeable box, as it confines the flow, while the region 0 < x , y , z < 2n is termed the periodicity box, as it reflects the periodicity of the Fourier series (1.2). Also, because of the symmetries listed in appendix A, the flow at any point in space can be inferred from its values in the fundamental box 0 < x, y, z < in.Secondly, if near each face we write the velocity field in terms of components parallel or perpendicular to the face, i.e. u = u,, + vl, then ul and au,,/an vanish on the face. This implies that the vorticity on each face is normal to that face so it may be written w = @, where fi is the unit normal. Note that g must vanish on all edges of the box where faces meet. I n contrast, a general incompressible flow will have only isolated points of vanishing vorticity. (Both velocity and vorticity also vanish for all time a t the centre x = y = z = in.)Thirdly, the vanishing of ul and au,,/an on each face also implies that the tensor V v is partly diagonal. One principal axis of the strain rate or symmetric part of this tensor is then perpendicular to the face. Furthermore, the magnitude of the strain rate along this axis determines the fractional growth rate of the normal vorticity on the face, i.e. d % = --~r * v --lnlcl. an ' Idt (2.1) 14-2

Quantum Hall states at filling fraction ν=5/2 are examined by numerical diagonalization. Spinpolarized and -unpolarized states of systems with N ≤18 electrons are studied, neglecting effects of Landau level mixing. We find that the ground state is spin polarized. It is incompressible and has a large overlap with paired states like the Pfaffian. For a given sample, the energy gap is about 11 times smaller than at ν=1/3. Evidence is presented of phase transitions to compressible states, driven by the interaction strength at short distance. A reinterpretation of experiments is suggested. 73.40.Hm,02.60.Dc, Ten years after the discovery of a quantized Hall plateau at filling fraction ν=5/2 by Willett et al. [1], 'a key piece of the ν=5/2 puzzle is still missing': This is the conclusion reached by Eisenstein in his recent review [2,3]. Studies by Eisenstein et al. [4] in a tilted magnetic field had shown that the plateau disappears when the tilt angle exceeds a critical value. It is now widely believed that the plateau is the result of a spin-unpolarized incompressible ground state (GS), while, at larger tilt angles, the Zeeman energy favors a polarized compressible GS, consistent with the disappearance of the plateau.The evidence supporting the above picture is taken from activation studies which reveal an energy gap that decreases with increasing tilt angle [5]. This fact is explained naturally if the GS is unpolarized and if its lowest energy excitations involve electrons with reversed spin, and thus a gain in Zeeman energy ∆E=gµ B B from spinreversal (g and µ B stand for the g-factor and the Bohr magneton). This energy gain increases with increasing tilt angle Θ as the magnetic field perpendicular to the sample, B ⊥ =BcosΘ, is fixed by the electron density n S of the sample and the filling fraction ν [6]. From the slope of the activation energy as a function of B, a gfactor g≈0.56 was extracted [5,3], somewhat larger than its value g=0.44 for bulk GaAs. That the polarized state expected at large tilt angles should be compressible, is consistent with the Fermion Chern-Simons theory of Halperin, Lee and Read [7], which predicts that electrons in a half-filled Landau level (LL) behave like quasiparticles in zero magnetic field forming a Fermi liquid, the 'Composite Fermion (CF) liquid ' [8].In this note, we challenge this interpretation of the experiments. We present evidence from exact diagonalization results that the GS in a half-filled second LL is spin-polarized and incompressible, consistent with the prediction by d'Ambrumenil and the author [9] that the CF-liquid does not form at this filling.What makes the plateau disappear at large tilt angles? If the system is spin-polarized already at small tilt angles, the Zeeman energy cannot drive the phase transition. In this note, we show that the incompressible state is very sensitive to details of the interaction: phase transitions to gapless states occur when the interaction at short distance is either 'too hard' or 'too soft'. When it is 'too hard', we recover the...

Diffractive optical structures for increasing the efficiency of crystalline silicon solar cells are discussed. As a consequence of the indirect band gap, light absorption becomes very ineffective near the band edge. This can be remedied by use of optimized diffraction gratings that lead to light trapping. We present blazed gratings that increase the optically effective cell thickness by approximately a factor of 5. In addition we present a wideband antireflection structure for glass that consists of a diffraction grating with a dielectric overcoat, which leads to an average reflection of less than 0.6% in the wavelength range between 300 and 2100 nm.

We compute energy gaps for spin-polarized fractional quantum Hall states in the lowest Landau level at filling fractions ν = 1 3 , 2 5 , 3 7 and 4 9 using exact diagonalization of systems with up to 16 particles and extrapolation to the infinite system-size limit. The gaps calculated for a pure Coulomb interaction and ignoring finite width effects, disorder and LL mixing agree well with the predictions of composite fermion theory provided the logarithmic corrections to the effective mass are included. This is in contrast with previous estimates, which, as we show, overestimated the gaps at ν =2/5 and 3/7 by around 15%. We also study the reduction of the gaps as a result of the non-zero width of the 2D layer. We show that these effects are accurately accounted for using either Gaussian or 'z× Gaussian' (zG) trial wavefunctions, which we show are significantly better variational wavefunctions than the Fang-Howard wavefunction. The Gaussian and zG wavefunctions give Haldane pseudopotential parameters which agree with those of self-consistent LDA calculations to better than ± 0.2%. For quantum well parameters typical of heterostructure samples, we find gap reductions of around 20%. The experimental gaps, after accounting heuristically for disorder, are still around 40% smaller than the computed gaps. However, for the case of tetracene layers in metal-insulator-semiconductor (MIS) devices we find that the measured activation gaps are close to those we compute. We discuss possible reasons why the difference between computed and measured activation gaps is larger in GaAs heterostructures than MIS devices. Finally, we present new calculations using systems with up to 18 electrons of the gap at ν = 5 2 including width corrections.

Monte Carlo methods have been employed to evaluate the energy of two previously proposed trial wave functions for the quasiparticle at the v= 3 quantized Hall state of the two-dimensional electron system. The two wave functions have the same energy within our statistical accuracy, and are consistent with a value e+( 3 )=(0.073%0.008)et/el"where io is the insgnetic length, and e the background dielectric constant. Simulations of the quasihole state confirm previous estimates of e (T)=0.026e /elo. We have also studied the charge distributions of the quasiparticle and quasihole states, and we have evaluated the energies of a previously proposed microscopic trial wave function for the ground state at v= -, , 3, and 7.